"THE IDEAL-GAS EQUATION OF STATE"
THE IDEAL-GAS EQUATION OF STATE • Equation of state: Any equation that relates the pressure, temperature, and specific volume of a substance. • The simplest and best-known equation of state for substances in the gas phase is the ideal-gas equation of state. This equation predicts the P-v-T behavior of a gas quite accurately within some properly selected region. Ideal gas equation of state Different substances have different gas constants. 1 Mass = Molar mass Mole number Ideal gas equation at two states for a fixed mass Real gases behave as an ideal Various gas at low NR NR T expressions densities (i.e., low of ideal gas pressure, high equation temperature). RT The ideal-gas relation often is not applicable to real gases; thus, care should be exercised when using it. Properties per unit mole are denoted with a bar on the top. 2 Is Water Vapor an Ideal Gas? • At pressures below 10 kPa, water vapor can be treated as an ideal gas, regardless of its temperature, with negligible error (less than 0.1 percent). • At higher pressures, however, the ideal gas assumption yields unacceptable errors, particularly in the vicinity of the critical point and the saturated vapor line. • In air-conditioning applications, the water vapor in the air can be treated as an ideal gas. Why? • In steam power plant applications, however, the pressures involved are usually very high; therefore, ideal-gas relations should not be used. Percentage of error ([|vtable - videal|/vtable] 100) involved in assuming steam to be an ideal gas, and the region where steam 3 can be treated as an ideal gas with less than 1 percent error. COMPRESSIBILITY FACTOR—A MEASURE OF DEVIATION FROM IDEAL-GAS BEHAVIOR Compressibility factor Z The farther away Z is from unity, the more the A factor that accounts for gas deviates from ideal-gas behavior. the deviation of real gases Gases behave as an ideal gas at low densities from ideal-gas behavior at (i.e., low pressure, high temperature). a given temperature and Question: What is the criteria for low pressure pressure. and high temperature? Answer: The pressure or temperature of a gas is high or low relative to its critical temperature or pressure. At very low pressures, all gases approach The compressibility factor is ideal-gas behavior (regardless of their unity for ideal gases. temperature). 4 Reduced Reduced pressure temperature Pseudo-reduced Z can also be determined from specific volume a knowledge of PR and vR. Gases deviate from the ideal-gas behavior the most in the neighborhood of the critical point. 5 Comparison of Z factors for various gases. Ideal Gas Model pv RT pR 1, TR 1 u u (T ) only h h(T ) u (T ) RT Do not be confused with superheated water vapor! INTERNAL ENERGY, ENTHALPY, AND SPECIFIC HEATS OF IDEAL GASES Joule showed using this experimental Internal energy and apparatus that For ideal gases, enthalpy change of u=u(T) u, h, cv, and cp an ideal gas vary with temperature only. 7 • At low pressures, all real gases approach • u and h data for a number of ideal-gas behavior, and therefore their gases have been tabulated. specific heats depend on temperature only. • These tables are obtained by • The specific heats of real gases at low choosing an arbitrary reference pressures are called ideal-gas specific point and performing the heats, or zero-pressure specific heats, and integrations by treating state 1 are often denoted cp0 and cv0. as the reference state. Ideal-gas constant- pressure specific heats for some gases (see In the preparation of ideal-gas Table A–2c tables, 0 K is chosen as the for cp reference temperature. 8 equations). Internal energy and enthalpy change when specific heat is taken constant at an average value (kJ/kg) For small temperature intervals, the specific heats may be assumed to vary linearly with temperature. The relation u = cv T is valid for any kind of process, constant- volume or not. 9 Three ways of calculating u and h 1. By using the tabulated u and h data. This is the easiest and most accurate way when tables are readily available. 2. By using the cv or cp relations (Table A-2c) as a function of temperature and performing the integrations. This is very inconvenient for hand calculations but quite desirable for computerized calculations. The results obtained are very accurate. 3. By using average specific heats. This is very simple and certainly very convenient when property tables are not available. The results obtained are reasonably accurate if the Three ways of calculating u. temperature interval is not very large. 10 Specific Heat Relations of Ideal Gases The relationship between cp, cv and R dh = cpdT and du = cvdT On a molar basis Specific heat ratio • The specific ratio varies with temperature, but this variation is very mild. • For monatomic gases (helium, argon, etc.), its value is essentially constant at 1.667. The cp of an ideal gas can be determined from a knowledge of • Many diatomic gases, including air, cv and R. have a specific heat ratio of about 1.4 at room temperature. 11 Internal Energy Changes of incompressible substance Enthalpy Changes The enthalpy of a compressed liquid A more accurate relation than 12 EX 3.8 One pound of air in a piston-cylinder assembly undergoes a thermodynamic cycle consisting of three processes. Process 1–2: Constant specific volume Process 2–3: Constant-temperature expansion Process 3–1: Constant-pressure compression At state 1, the temperature is 540°R, and the pressure is 1 atm. At state 2, the pressure is 2 atm. Employing the ideal gas equation of state, (a) sketch the cycle on p–v coordinates. (b) determine the temperature at state 2, in °R. (c) determine the specific volume at state 3, in ft3/lb. Power cycle or Refrigeration cycle? Using Ideal Gas Tables A piston-cylinder assembly contains 2 lb of air at a temperature of 540 oR and a pressure of 1 atm. The air compressed to a state where the temperature is 840 oR and the pressure is 6 atm. During the compression, there is a heat transfer from the air to the surroundings equal to 20 Btu. Using the ideal gas model for air, determine the work during the process in Btu. 690 oR = 230 oF Specific heat functions cp T T 2 T 3 T 4 h2-h1? R Using constant c u (T2 ) u(T1 ) cv (T2 T1 ) h(T2 ) h(T1 ) c p (T2 T1 ) Two tanks are connected by a valve. One tank contains 2 kg of carbon monoxide gas at 77 oC and 0.7 bar. The other tank holds 8 kg of the same gas at 27 oC and 1.2 bar. The valve is opened and the gases are allowed to mix while receiving energy by heat transfer from the surroundings. The final equilibirium temperature is 42 oC. Using the ideal gas model with constant cv, determine (a) the final equilibrium pressure, in bar (b) the heat transfer for the process, in kJ. Polytropic, Isothermal, and Isobaric processes Polytropic process: C, n (polytropic exponent) constants Polytropic process n 1 n 1 T2 V1 P n Polytropic and for ideal gas 2 T1 V2 P 1 When n = 1 (isothermal process) Constant pressure process What is the boundary work for a constant- volume process? Schematic and P-V diagram for a polytropic 19 process. Ex. 3.12 • Air undergoes a polytropic compression in a piston–cylinder assembly from p1=1 atm, T1=70 oF to p2=5 atm. Employing the ideal gas model with constant specific heat ratio k, determine the work and heat transfer per unit mass, in Btu/lb, if (a) n=1.3, (b) n=k . Evaluate k at T1. • 3.115 One kilogram of nitrogen fills the cylinder of a piston-cylinder assembly, as shown in Fig. P3.115. There is no friction between the piston and the cylinder walls, and the surroundings are at 1 atm. The initial volume and pressure in the cylinder are 1 m3 and 1 atm, respectively. Heat transfer to the nitrogen occurs until the volume is doubled. Determine the heat transfer for the process, in kJ, assuming the specific heat ratio is constant, k=1.4. 3.135 A piston-cylinder assembly contains air modeled as an ideal gas with a constant specific heat ratio, k=1.4. The air undergoes a power cycle consisting of four processes in series: Process 1–2: Constant-temperature expansion at 600 K from p1=0.5 MPa to p2=0.4 MPa. Process 2–3: Polytropic expansion with n=k to p3=0.3 MPa. Process 3–4: Constant-pressure compression to V4=V1. Process 4–1: Constant-volume heating. Sketch the cycle on a p–v diagram. Determine (a) the work and heat transfer for each process, in kJ/kg, and (d) the thermal efficiency.